EffectiveSupercell

To correctly evaluate the Coulomb integrals when dealing with wavefunctions at different k-points, we need to consider the integral over a volume in which the integrals over the exponential factors vanish (the proof of this is attached as a pdf. The tex file is also around if needed).

This leads us to the concept of an effective supercell, which is the non-primitive cell that is exactly equivalent to the calculation over the k-point mesh.

As a result, we are restricted in choice of the k-point mesh to those which are physically meaningful: each component of each wavevector must be of the form 1/n, where n is an integer (units: reciprocal lattice basis). In order to sample each replication of the primitive cell in the effective supercell, the k-point mesh must be regularly spaced, with each k-point separated from the nearest along each axis by <math>1/n_i</math>, where <math>n_i</math> is the number of k-points along axis <math>i</math>. We also require the <math>\Gamma</math>-point to be included. Such meshes are easy to generate, although somewhat tedious. A simple python script (~jss43/bin/kpntgen.py) on sword generates the desired mesh for inclusion in CPMD input files.

We can apply (with some care) symmetry operations to our wavefunctions expanded out into the effective supercell.

Supercells (both in <math>\Gamma</math>-point calculations and k-point calculations) have a subtle problem with symmetry in that symmetry operations of the lattice are not necessarily symmetry operations of the supercell (and hence of the wavefunctions), if the supercell is not an <math>n\times n\times n</math> enlargement of the primitive cell. For a symmetry operation of the lattice to be a symmetry operation of the cell, we require the length of each lattice vector to be unchanged by the operation. This amounts to considering the effect of the operation on the Cartesian basis vectors. Hence we require the columns of the rotation matrix to be of unit magnitude in order for the operation to be a symmetry operation of the cell. --james 11:41, 5 February 2007 (GMT)